MSc Physics
Theoretical Physics
M
ASTERT
HESISInvestigating the ER=EPR proposal
A field theory study of long wormholes
by Bernardo Zan 10513752 May 2015 54 ECTS Supervisor:
Prof. dr. Jan de Boer
Examiner: dr. Ben Freivogel
The ER=EPR correspondence states that two entangled systems are always connected by an Einstein Rosen bridge. This proposal might solve the question of black hole evaporation with-out requiring the presence of firewalls. However, it has been argued that the conditions nec-essary to have a smooth geometrical description cannot be satisfied by the randomness of a typical state. It was shown that correlators between two entangled theories are weak for typ-ical states and depend on the spectrum of the field theory; this is taken as evidence for the absence of a semiclassical wormhole. We study a non typical state which presents a long semi-classical wormhole description from the field theory point of view. We find that, under certain conditions, a random matrix approach is able to model this state: the correlator between the two theories is small and it does not depend on the spectrum of the theories.
Introduction 1
1 Quantum field theory at finite temperature 5
1.1 KMS condition and periodicity of fields . . . 5
1.1.1 Path integral for the partition function. . . 6
1.2 Thermofield dynamics . . . 7
1.3 Real time formalism . . . 8
1.4 Schwinger Keldysh propagators . . . 9
1.5 Free bosonic field . . . 11
1.6 Bosonic propagators in position space . . . 12
1.6.1 Operators on the same theory . . . 13
1.6.2 Operators on different theories . . . 15
1.6.3 Massless limit . . . 16
2 The ER=EPR proposal 19 2.1 AdS space and the AdS/CFT correspondence . . . 19
2.2 The BTZ black hole . . . 20
2.2.1 Propagators for the BTZ black hole . . . 22
2.2.2 Holography for the BTZ black hole. . . 23
2.3 Hawking radiation . . . 24
2.4 The information paradox and the firewall proposal . . . 26
2.5 The ER=EPR proposal . . . 28
2.5.1 Reactions to the ER=EPR proposal . . . 31
2.6 Long semiclassical wormholes . . . 32
2.7 Motivation . . . 35
3 Quenching the system 37 3.1 Response theory to second order . . . 37
3.2 A first toy model . . . 38
3.3 A second toy model . . . 39
3.3.1 On a circle . . . 40
4 Random matrices 43
4.1 Why random matrices? . . . 43
4.2 Ensembles and states . . . 44
4.2.1 Non energy changing operators . . . 45
4.2.2 Energy changing operators . . . 45
4.3 Random matrices and wormholes . . . 46
4.4 Random matrices and time dependence . . . 48
4.5 Unitary matrices. . . 49
4.6 Numerical results . . . 52
4.6.1 Triangular matrices . . . 53
4.6.2 Matrices with no diagonal elements . . . 53
4.7 Comparison with the bulk description . . . 56
Conclusion 59 Appendices 61 A Second order quench 61 A.1 On the line . . . 61
A.2 On the circle . . . 63
B Weingarten function 65 B.1 Weingarten function . . . 65
B.2 Unitary matrices on the microcanonical state . . . 66
B.3 Unitary matrices on the TFD state. . . 66
Bibliography 69
Many aspects of the physics of black hole are still not understood or subject to debate. Since the discovery of Hawking radiation [1] in 1975, combining gravitation and quantum effects has proven to be a challenging task. The fate of the information that enters a black hole after this evaporates through Hawking radiation was initially not well understood. This radiation appears thermal to a distant static observer. Unitarity, one of the well established foundations of quantum mechanics, appeared to be violated by the evolution of a black hole from a pure state to such a mixed state. Most of the scientific community agrees now that the evaporation of a black hole is a unitary process. The information leaks out through entanglement between the quanta being emitted through Hawking radiation at different times.
The AdS/CFT correspondence [2] has proven important in answering this question. This correspondence relates a gravitational theory on a manifold which is asymptotically AdS, a spacetime with a negative cosmological constant, in d+1 dimensions to a conformal field the-ory, a quantum field theory that is conformally invariant, living on the d dimensional boundary of the manifold. When one of the two theories is strongly coupled, the other one is weakly cou-pled. This correspondence has many applications. One of these, for example, is to carry out computations in strongly coupled theories that would not be possible otherwise. It turns out to be useful also when considering the process of evaporation of a black hole. Non unitary of this process would not be consistent with the AdS/CFT correspondence. Since some types of black holes are dual to conformal field theories which evolve unitarily, their evaporation must be unitary as well.
However, the solution of the information paradox has generated other apparent inconsis-tencies. The quanta emitted through Hawking radiation are entangled with quanta inside the black hole. Information starts leaking out after half of the black hole has evaporated. If effec-tive field theory is a valid description of the physics outside the horizon, this information must be carried by the entanglement between early and late Hawking radiation. These modes are approximately maximally entangled: monogamy of entanglement, the principle for which one system cannot be maximally entangled with two independent systems, is violated. In 2012 it was proposed [3] that the modes of Hawking radiation are not entangled with modes inside the black hole, and monogamy of entanglement is not violated. However, there is not a clear mechanism which would allow this breaking of entanglement. Besides, it would mean that just behind the horizon there are many energetic particles, reason for which this proposal is called
Introduction 2
the “firewall” proposal. The horizon of the black hole would then be a special place for a freely falling observer, hence violating Einstein equivalence principle.
In 2013 a different solution of the paradox was proposed [4]. It conjectures a correspondence between entanglement and Einsten Rosen bridges, and goes under the name of ER=EPR. Ac-cording to the proposal, two entangled systems are always connected by a wormhole. There are cases for which this is known to be true. The BTZ black hole [5] is dual to an entangled state, the thermofield double state (TFD) [6]. However, according to the ER=EPR proposal, the duality holds for all entangled systems. For example, even simpler systems, such as two entan-gled qubits, should be connected by a wormhole; however, this wormhole will have a highly quantum nature, and cannot be studied given our current understanding of quantum gravity.
If correct, the ER=EPR proposal could imply the absence of firewalls at the black hole hori-zon, consistently with Einstein equivalence principle. An observer trying to measure the vio-lation of the monogamy of entanglement would first measure a mode of the early and one of the late Hawking radiation. Since information is not lost when a black hole evaporates, the observer would see that these two modes are entangled. Then he would enter the black hole to check the entanglement between the late radiation mode and its Hawking partner mode in-side the black hole. He would however meet an energetic particle on the horizon, making the measurement impossible. This energetic particle would be created by the action of the observer when he first made a measurement on the early Hawking radiation. This action at a distance would be possible only because, according to the ER=EPR proposal, two entangled systems are connected by a wormhole. This small violation of locality might explain the process of evaporation of black holes without requiring the presence of firewalls.
Investigating the ER=EPR proposal turns out to be a difficult task. Different works [7,8] have argued that, while the TFD is dual to a semiclassical wormhole, typical states are not. Correlations between the two entangled systems suppressed by factors of e−S, dependent on the spectrum of the field theory, are taken as evidence of the absence of a wormhole, since the geometry should arise from the coarse graining of the field theory. However, it was shown in [9] and [10] by Shenker and Stanford that it is possible to construct long semiclassical wormholes, for which correlators can be made of order e−S.
In [8] random matrices were used to study states more general than the TFD. Under the assumption that a correlator that depends on the number of states of a theory is evidence for the absence of a wormhole, the authors claimed that typical states are not connected by wormholes. The SS construction is, instead, highly non typical. The main questions we will try to answer in this work concerns the validity of a free field or a random matrix approach for this latter SS state and whether this state presents correlations that depend on the number of sates of the field theory. If e−Scorrelations follow only from an ad hoc construction, it would look like the claim of [7,8] is correct and typical states are not dual to smooth geometries.
The structure of this work is as follows. In chapter1we review some properties of quan-tum field theory at finite temperature, and focus on the bosonic 1+1 dimensional case. This will play a role in the context of the AdS/CFT correspondence. In chapter 2, the BTZ black hole, which provides an example of a wormhole with an entangled dual, is reviewed. We also
explain the process of black hole evaporation and the firewall as well as the ER=EPR proposal. In chapter3we try to model the SS construction, which consists in introducing matter on the boundary of the black hole, using a quench in a free field theory. Eventually, in chapter4, we try to model the same process using random matrices drawn from different ensembles.
1
Quantum field theory at finite
temperature
Quantum field theory was initially developed at zero temperature; much work has been de-voted to generalizing its results to finite temperature. Different approaches, allowing to keep some tools of T =0 QFT, exist; in the following section we will review the thermofield formal-ism and the real time formalformal-ism. The first makes it possible to express the statistical value of an operator as the expectation value of the operator in a quantum field theory, and has a direct application when talking about black holes in the context of the AdS/CFT correspondence; the second achieves the generalization of Feynman diagrams to finite temperature. We will show that the latter approach is equivalent to the first one, and proves useful in computations in the same context. Later, we will specialize in the case of a free boson in 1+1 dimensions, which will be useful in the context of the AdS/CFT correspondence, as will be explained in chapter2. We begin by reviewing some general features of finite temperature field theory. At finite temperature T, the expectation value of an operator O is given by
hOi =Tr(ρeqO) (1.1)
where ρeq is the thermal density matrix, ρeq = e−βH/Z and β = T1 (in this work we set
Boltz-mann’s constant kBto 1). This density matrix represents the case of a canonical ensemble. In the
presence of a Noether conserved charge Q and a chemical potential µ, we need to consider the grand canonical density matrix, ρ= 1
Ze−β(H+µQ). However, for our purposes it will be enough
to consider the canonical ensemble.
1.1 KMS condition and periodicity of fields
Lorentz invariance is broken at finite temperature. It is possible to see this from the Kubo-Martin-Schwinger (KMS) condition.
If we consider the expectation value of two operatorshA(t)B(t0)iat finite temperature, we 5
KMS condition and periodicity of fields 6
can use the cyclicity of the trace to see Tr(ρeqA(t)B(t0)) = 1 ZTr(e −βHA(t)B(t0)) = 1 ZTr(e −βHB(t0)e−βHA(t)eβH) = 1 ZTr(e −βHB(t0)A(t+iβ)). (1.2)
We are working in the Heisenberg picture, A(t) = eitHA(0)e−itH. We have therefore what is
known as the KMS condition:
hA(t)B(t0)i = hB(t0)A(t+iβ)i (1.3) If we define (expressing only the time dependence for simplicity) C>(t−t0) = C<(t0−t) = hφ(t)φ(t0)i, then we have
C>(t−iβ) =hφ(t−iβ)φ(0)i = hφ(0)φ(t)i =C<(t). (1.4)
Fourier transforming the two sides of (1.4) we obtain
hφkφ−ki =eβk0hφ−kφki, (1.5)
and we see that, because of the term eβk0, this is not Lorentz invariant. An equivalent way of stating that finite temperature breaks Lorentz invariance is that, when inserting a system in a heat bath, we have a preferred frame, the rest frame of the heat bath.
1.1.1 Path integral for the partition function
The partition functionZ is defined as the trace of the density matrix
Z =Tr ρeq. (1.6)
This quantity plays a crucial role in statistical mechanics, since most of the interesting thermo-dynamic quantities can be obtained from it.
It has been long known that statistical quantum mechanics is connected to quantum field theory in Euclidean space. In fact, it is possible to express the partition functionZ as a path integral in Euclidean space, with some periodic boundary conditions.
The lagrangian for a real scalar field φ is L =−1
2∂µφ∂ µ
φ−V(φ) (1.7)
where V(φ)is a potential. The transition amplitude can be expressed as (see for example [11]) hφb(x)|e−iHt|φa(x)i = Z Dφ exp i Z t 0 dt Z dd−1xL(x) (1.8)
with the condition φ(x0 =0, x) =φa(x)and φ(x0 =t, x) =φb(x). We can use this to compute
the partition function
Z =Tr e−βH= Z Dφhφ|e−βH|φi (1.9) so that Z = N Z Dφ exp i Z −iβ 0 dt Z dd−1xL[φ] . (1.10)
where the fields φ satisfy the periodic condition φ(0) =φ(−iβ). It is convenient now to go to
Euclidean time, performing the change of variable τ =it. This gives Z = N Z periodicDφ exp − Z β 0 dτ Z dd−1xLE[φ] (1.11)
whereLE = 12(∂τφ)2+12(∂iφ)2+V(φ)is the Euclidean lagrangian density. The identification
of imaginary time τ = τ+βshows as well how Lorentz invariance is broken at finite
temper-ature.
1.2 Thermofield dynamics
Thermofield dynamics was developed in [12]. Its aim is to express statistical averages at a finite temperature as the expectations value in a quantum field theory. The idea is to find a temperature dependent state|Ψβifor which
hΨβ|A|Ψβi = Tr(e−βHA) Tr(e−βH) = 1 Z
∑
n e −βEnhn|A|ni (1.12)It is not possible to create such a state using only one Hilbert space. However, if we consider another (fictitious) Hilbert space, we can build the following state, which will hold the desired property. |Ψβi = 1 √ Z
∑
n e−βEn/2|ni ⊗ |˜ni (1.13) where Z = ∑ne−βEn. This is usually called thermofield double state (TFD), and we will write it as|Ψi, dropping the β dependence.The state|nibelongs to the first Hilbert space, while|˜nito the doubled Hilbert space. The usual relations h˜n|m˜i = δnm hold for the latter states as well. If we compute the expectation
value of an operator living in the first Hilbert space, then we find the required relation (1.13). This is because, if we start from the density matrix ρ = |Ψi hΨ|, the reduced density matrix obtained by tracing out the second Hilbert space is the thermal density matrix ρeq =e−βH/Z.
It is also possible to consider operators ˜O living in the second Hilbert space. Inside the context of thermofield dynamics it might not be immediately clear what these fictitious opera-tors represent physically. However, it will be explained in chapter2that in the context of the AdS/CFT correspondence these operators have a straightforward interpretation.
Real time formalism 8 ti O1(t) tf O2(t0) tf −iσ ti−iσ ti−iβ Im t Re t
Figure 1.1: Schwinger Keldysh time contour.
have a Lagrangian formulation we turn to the real time formalism, which we will show to be equivalent to the thermofield dynamics approach.
1.3 Real time formalism
Another formalism allowing the computation of expectation values of physical observables at finite temperature is the real time, or time path, formalism. This method was developed by Schwinger and Keldysh [13, 14] and it allows the generalization of Feynman rules to finite temperature.
Similarly to the previous approach, this formalism doubles the degrees of freedom: besides the ordinary physical fields, living at real times, there are also ghost fields which live at imagi-nary times.
The time path is shown in figure1.1. It extends into the complex plane: it runs on the real axis from a time ti to a time tf, then runs in the imaginary time direction to tf −iσ, runs back
parallel to the real axis to ti−iσ1, and eventually runs downward to ti−iβ. This latter point
is identified with ti, since the fields are periodic in imaginary time with period β. The limits
ti → −∞ and tf → ∞ are taken. Boundary conditions are imposed so that fields vanish at
t=±∞, and the contributions of the vertical segments of the contour vanish.
Operators living on the real time contour are denoted as O1, while the ghost operators on
the imaginary time segment are denoted as O2. They are defined as
O1(t) =O(t) =eiHtOe−iHt
O2(t) =O(t−iσ) =eiH(t−iσ)Oe−iH(t−iσ)
(1.14)
While σ can take any value, provided 0 < σ < β, the most convenient choice in our case
is to take σ = β/2. The reason for this is that such a symmetric choice makes this formalism
closely related to the TFD state, and it will be important for our future applications.
This approach is equivalent to the thermofield formalism. In order to see this, we set σ =
1Actually, the horizontal segments are slightly tilted in order to have the correct pole prescription. The upper
β/2 and we look for an operator ˜B acting on the second Hilbert space such that
Tr(ρeqA(t)B(t0+iβ/2)) =hΨ|A(t)B˜(t0)|Ψi. (1.15)
Remembering that B(t0+iβ/2) =e−βH/2B(t0)eβH/2, the previous equation becomes
1 Z
∑
n,me −βEn/2e−βEm/2hn|A(t)|mi hm|B(t0)|ni = = 1 Zn,m∑
e −βEn/2e−βEm/2hn|A(t)|mi hn|B˜(t0)|mi. (1.16)From this equation we can see that we require ˜B = BT. Therefore an operator living on the second Hilbert space in the thermofield formalism is related to an operator living on the imag-inary part of the Schwinger-Keldysh contour. The previous formula can be generalized to the insertion of many operators B in the same manner.
1.4 Schwinger Keldysh propagators
The TFD state appears in the context of the AdS/CFT correspondance. We will be interested in computing the propagator between the two different Hilbert space; having shown that the TFD formalism is the same as the real time formalism, we turn to this last one to compute the propagators.
We define the Schwinger Keldysh propagators as
iD11(x, t) =hTO1(x, t)O1(0)i
iD12(x, t) =hO2(0)O1(x, t)i
iD21(x, t) =hO2(x, t)O1(0)i
iD22(x, t) =hTO¯ 2(x, t)O2(0)i.
(1.17)
T denotes time ordering, while ¯T denotes anti time ordering (the second branch of the Schwinger Keldysh time contour has a different pole prescription from that of the first branch). We define also the retarded propagator
iDR(x, t) =θ(t)h[O1(x, t), O1(0)]i. (1.18)
This turns out to be a useful quantity, since it allows us to express the propagators (1.17) in momentum space. Besides, we will later focus on a free boson. For this theory, the retarded propagator is state independent, therefore it is possible to compute it at T =0, where Lorentz invariance is still present. Omitting for the moment any spatial coordinate, which remains trivial, we can express D11as
iD11(t) = 1 Z
∑
a ha|e −βH θ(t)O1(t)O1(0) +θ(−t)O1(0)O1(t) |ai. (1.19)Schwinger Keldysh propagators 10
Inserting a complete set of states and expressing O1(t) =eiHtO(0)e−iHtwe find
iD11(t) =
1 Z
∑
a,bei(Ea−Eb)tO
abOba θ(t)e−βEa+θ(−t)e−βEb. (1.20)
We now Fourier transform to momentum space iD11(ω) =
Z ∞
−∞iD11(t)e
iωt−ε|t|. (1.21)
The ε factor makes the integral convergent. After solving the integral we find D11(ω) = 1 Z
∑
a,bOabOba e−βEa ω+ (Ea−Eb) +iε − e−βEb ω+ (Ea−Eb)−iε . (1.22)The same procedure can be applied to DR. We find
iDR(t) = θ(t)
Z
∑
a,bei(Ea−Eb)tO
abOba(e−βEa−e−βEb) (1.23)
and, in momentum space, DR(ω) = 1 Z
∑
a,bOabOba e−βEa ω+ (Ea−Eb) +iε− e−βEb ω+ (Ea−Eb) +iε . (1.24)Using the Dirac identity
lim ε→0+ 1 x∓iε = P 1 x±iπδ(x), (1.25) we can find the following relations
Re D11(ω) =Re DR(ω) (1.26) Im D11(ω) = e βω+1 eβω−1Im D R( ω) =cothβω 2 Im D R( ω). (1.27)
Repeating the same for D21, we find
D12(ω) =−1 Z
∑
a,be −βEa/2e−βEb/2O abOba 1 w+ (Eb−Ea)−iε − 1 w+ (Eb−Ea) +iε (1.28) and therefore D12(ω) = 2ie −βω/2 1−e−βωIm D R( ω). (1.29)The same analysis can be repeated for D21and D22. D21 is the same as D12 because of the
symmetric choice σ= β/2, and D22has the same imaginary part as D11but opposite real part,
due to the reverse time ordering ¯T
D22(ω) =−D11(ω)∗ =−Re DR(ω) +i cothβω
2 Im D
R(
ω) (1.31)
The Schwinger Keldysh formalism makes it possible to generalize Feynman rules to finite temperature. The propagator for the field φ is now represented by the 2×2 matrix Dij. Since the
field living on imaginary times φ2plays the role of an unphysical or ghost field, the external legs
of all Feynman diagrams will be the physical field φ1. The interaction between the two types of
fields is possible thanks to the off diagonal elements of the matrix D, and in the internal lines of the Feynman diagram both φ1and φ2will appear. There are two types of interaction vertex,
one for each kind of field.
In this work we will treat free theory at finite temperature, therefore we will not use Feny-man diagrams. We are only interested in the Dij matrix in the bosonic case. A more detailed
overview of Fenyman diagrams in the Schwinger Keldysh formalism, as well as a treatment of the fermionic case, can be found in [15].
1.5 Free bosonic field
Given the lagrangian for a free bosonic field in d dimensions L =−1 2∂µφ∂ µ φ−1 2m 2 φ2 (1.32)
the mode expansion for the field is
φ(x, t) = Z dd−1k (2π)d−1 1 2ωk [akeik·x+a†ke−ik·x] (1.33) where ωk = √
k2+m2. In thermal equilibrium we have [16]
haka†k0i = (2π)d−12ωk[N(ωk) +1]δ(k−k0) ha†kak0i = (2π)d−12ωkN(ωk)δ(k−k0)
(1.34)
with N(ωk) = (eβωk−1)−1.
We define C>(x, y) =C<(y, x) =hφ(x)φ(y)i, and the spectral density as
ρ(k) =C>(k)−C<(k). (1.35)
The spectral density turns out to be an useful quantity, since it allows us to find the retarded and advanced propagator using the relations
DR(k0, k) = Z ∞ −∞ dk00 2π ρ(k00, k) k00−k0−iε DA(k0, k) = Z ∞ −∞ dk00 2π ρ(k00, k) k00−k0+iε . (1.36)
Bosonic propagators in position space 12
In the case of a free bosonic field we find
ρ(k) = 2π 2ωk δ(k0−ωk)−δ(k0+ωk) = 2π sgn(k0)δ(k2+m2) (1.37) where sgn(k0) =θ(k0)−θ(−k0). We have DR(k) = 1 −(k0+iε)2+ωk2 (1.38)
This quantity is independent of the temperature. The temperature dependence appears only when we turn to the time ordered correlator.
Now, let’s consider the propagators Dabfor O =φ. Using
Re DR(k) =P 1 −k20+ωk2 Im DR(k) =πsgn(k0)δ(k2+m2) (1.39) we determine D11(k) = 1 k2+m2−iε+ 2πi eβ|k0|−1δ(k 2+m2) D12(k) =D21(k) = 2πie−β|k0|/2 1−e−β|k0| δ(k 2+m2) D22(k) =− 1 k2+m2−iε + 2πi eβ|k0|−1δ(k 2+m2). (1.40)
We see clearly that these quantities are not Lorentz invariant. If we take the limit T → 0, however, we find the usual correlator for a free boson. The off diagonal elements D12 and
D21 vanish at zero temperature, because the two theories in the TFD state are not entangled
anymore, or because the two horizontal parts of the Schwinger Keldysh contour are infinitely far apart. We find the usual rules for Feynman diagrams. Since the first and second field cannot interact anymore, and in the outer legs of the Feynman diagrams only the physical field
φ1 appears, the whole Feynman diagram will only consist φ1 and the ghost field φ2 will not
play a role.
1.6 Bosonic propagators in position space
Throughout this work we will be interested in the 1+1 dimensional case. We compute now the Fourier transform of the Schwinger Keldysh propagators.
Dab(x) =
Z d2k
(2π)2e ik·xD
1.6.1 Operators on the same theory
The first term of D11in (1.40) gives
1 (2π)2 Z dkdk0 eik·x −k20+ω2k−ie =− 1 (2π)2 Z dkeikx Z dk0 eik0t k20−ω2k+ie = i 2π Z dke ikx 2ωk θ(t)e− iωkt+θ(−t)eiωkt = i 4πθ(t) Z dk ωk eikx−iωkt+ t→ −t = i 4πθ(t) Z
dye−im(t cosh y−x sinh y)+ t→ −t.
(1.42)
We first carried on the usual contour integration around the two poles and then made the change of variables k= m sinh y.
= i 4πθ(t) Z dye−m√x2−t2cosh(y+a)+ t → −t = i 2πθ(t)K0(m p x2−t2) + t→ −t = i 2πK0(m p x2−t2). (1.43)
We used d cosh(y+a) = d cosh a cosh y+d sinh a sinh y = it cosh y−ix sinh y, to find that d=√x2−t2and a=tanh−1(−x/t). Eventually we made the shift y→y−a. This is the usual
Feynman correlator for T=0. The second term is
1 (2π)2 Z dkdk0eik·x 2πi eβ|k0|−1δ(k 2+m2) = i 2π Z dkdk 0 2ωk eik·x e− β|k0| 1−e−β|k0| δ(k0+ωk) +δ(k0−ωk) = i 2π Z dk 2ωk e−βωk 1−e−βωke
ikx eiωkt+e−iωkt.
(1.44)
Now, using the expansion 1−1x =∑ xn, we can express the integral as
i 2π ∞
∑
n=0 Z dk 2ωke−(n+1)βωkeikx eiωkt+e−iωkt
= i 4π ∞
∑
n=0 Zdye−m[(n+1)β+it]cosh y+ix sinh y+ t→ −t
(1.45)
where we made the substitution k = m sinh y. Now we repeat the same trick as before. Using dncosh(y+an) =dncosh y cosh an+dnsinh y sinh an, we can rewrite the previous integral as
i 4π ∞
∑
n=0 Z ∞ −∞dye −mdncosh(y+an) (1.46)Bosonic propagators in position space 14
where dnand anare given by
dncosh an = (n+1)β+it
dnsinh an =−ix
(1.47)
We find therefore the explicit values
d2n = ((n+1)β+it)2+x2
an =tanh−1 −ix (n+1)β+it
!
(1.48)
Since the imaginary part of anis given by
Im an= −Re tan−1
x
(n+1)β+it
!
, (1.49)
then Im an∈ [−π2,π2]. Since the integral does not present singularities, and Re cosh(x+ib) >0
if x∈R and b∈ [−π 2,
π
2], we can make the shift y→y−an = i 4π ∞
∑
n=0 Z dye−m√(β(n+1)+it)2+x2cosh y+ t→ −t = i 2π ∞∑
n=0 K0 m q (β(n+1) +it)2+x2+K0 m q (β(n+1)−it)2+x2 = i 2π −2∑
n=−∞ +∑
∞ n=0 K0 m q (β(n+1) +it)2+x2. (1.50)Summing the two terms together we obtain D11(x, t) = i 2π ∞
∑
n=−∞K0 m q (β(n+1) +it)2+x2 = i 2π ∞∑
n=−∞ K0 m q (βn+it)2+x2 . (1.51)Using the relation in momentum space D22(k) =−D11(k)∗, we can find D22(x)in positions
space: D22(x) = Z dk (2π)2(−D11(k))∗e ik·x =− Z dk (2π)2D11(k)e− ik·x ∗ =−D11(−x)∗ = D11(x) (1.52)
1.6.2 Operators on different theories
To find D12in position space we follow the same method.
D12(x, t) = i 2π Z dkdk0 e−β|k0|/2 1−e−β|k0|e −ik0t+ikx δ(k2+m2) = i 2π Z dkdk 0 2|k0| e−β|k0|/2 1−e−β|k0|e −ik0t+ikx[δ(k 0+ωk) +δ(k0−ωk)] = i 2π Z ∞ −∞ dk 2ωk e−βωk/2 1−e−βωke
+ikxe−iωkt+e+iωkt
(1.53)
Let us consider the first term only for the moment. Using the series expansion (1−x)−1 =
∑∞n=0xn i 2π Z ∞ −∞ dk 2ωk e−βωk/2 1−e−βωke +ikxe−iωkt = i 2π ∞
∑
n=0 Z ∞ −∞ dk 2ωk e−(n+12)βωke+ikxe−iωkt = i 4π ∞∑
n=0 Z ∞ −∞dye −((n+12)β+it)m cosh yeimx sinh y
(1.54)
where in the last line we have made the usual change of variables k= m sinh y, for which
ωk =m cosh y and dk=ωkdy.
Using again d cosh(y+a) = d cosh y cosh a+d sinh y sinh a, we can rewrite the previous integral as i 4π ∞
∑
n=0 Z ∞ −∞dye −mdncosh(y+an) (1.55) where dnand anare given this time byd2n= (n+ 1 2)β+it 2 +x2 an=tanh−1 − ix (n+1 2)β+it ! (1.56)
Once again, we have no problem with the shift y →y−an
i 4π ∞
∑
n=0 Z ∞ −∞dye −mdncosh(y)= i 2π ∞∑
n=0 K0 m s (n+ 1 2)β+it 2 +x2 ! . (1.57)Considering also the second part of the integral, we obtain
D12(x, t) = i 2π ∞
∑
n=0 K0 m s (n+1 2)β+it 2 +x2 ! + i 2π ∞∑
n=0 K0 m s (n+1 2)β−it 2 +x2 ! = i 2π ∞∑
n=−∞ K0 m s (n+ 1 2)β+it 2 +x2 ! . (1.58)Bosonic propagators in position space 16
We note that both D11 and D12 are periodic in t → t+iβ, as required by the periodic
identification of the complex time coordinate. We also note that D11(x, t+iβ2) =D12(x, t).
1.6.3 Massless limit
Since we will turn our attention to the AdS/CFT correspondence, we are interested in confor-mal field theories. A free CFT is given by the lagrangian (1.7) with m = 0, since a mass term would introduce a length scale that would make the theory not scale invariant. For m → 0, both D11and D12diverge. The behavior of the K0function, for small argument, is
K0(x)' −log( x 2)−γE+O(x 2) as x →0. (1.59) Therefore D011(x, t) = i 2π ∞
∑
n=−∞ " −γE+log 2−log(m r nβ+it2+x2 # = i 2π ∞∑
n=−∞ "−γE+log(2)−log(m)−1
2log nβ+it2+x2 # . (1.60)
We get rid of the−γE+log(2)−log(m)term, to obtain, as a massless propagator,
D011(x, t) =− i 4π ∞
∑
n=−∞ lognβ+it2+x2. (1.61)This sum clearly diverges, so we need to renormalize it. In order to do so we first derive twice with respect to the x coordinate; this will make the sum convergent. After the sum is taken care of, we integrate back. This gives
D011(x, t) =− i 4π log sinh π(t−x) β +log sinh π(t+x) β =− i 4πlog 1 2cosh 2πt β − 1 2cosh 2πx β . (1.62)
The same procedure can be carried out for D12, and we obtain as a final result
D012(x, t) =− i 4π log cosh π(t−x) β +log cosh π(t+x) β =− i 4πlog 1 2cosh 2πt β + 1 2cosh 2πx β . (1.63)
In a two dimensional conformal field theory, the field φ does not have a definite scaling dimension. Instead, the operator ∂φ has weight(1, 0), and ¯∂φ has weight(0, 1). ∂ and ¯∂ are the shorthand notation for
∂= 1
2(∂x−∂t) ¯∂= 1
Therefore for these fields we have the following propagators2 h∂φ1(x, t)∂φ1(0)i = π 4β2 1 [sinhπ β(x−t)] 2 h¯∂φ1(x, t)¯∂φ1(0)i = π 4β2 1 [sinhπ β(x+t)] 2 h∂φ1(x, t)∂φ2(0)i = π 4β2 1 [coshπ β(x−t)] 2 h¯∂φ1(x, t)¯∂φ2(0)i = π 4β2 1 [coshπ β(x+t)] 2 h∂φi(x, t)¯∂φj(0)i =0. (1.65)
Now let’s consider the operator Oi =:∂φi¯∂φi:, which has weight(1, 1). We have hO1(x, t)O1(0)i = π2 16β4 1 h 1 2cosh2πxβ − 1 2cosh2πtβ i2 hO1(x, t)O2(0)i = π2 16β4 1 h 1 2cosh2πxβ + 1 2cosh2πtβ i2. (1.66)
We can also consider the operator Vαj =:eiαφj:, which has weight(α
2
8π, α
2
8π). Sinceh:eiαφi(x,t): :eiαφj
(0): i = e−α2hφi(x,t)φj(0)iwe have hVα1(x, t)Vα1(0)i = 1 h 1 2cosh2πxβ − 1 2cosh2πtβ i4πα2 hVα1(x, t)Vα2(0)i = 1 h 1 2cosh2πxβ + 1 2cosh2πtβ iα24π . (1.67)
It is worth noting that these quantities can also be obtained by conformal mapping the plane onto the cylinder.
2There is a small sublety here: the (massive) field on the second boundary is quantized as φ
2(x, t) =
R dk
2π2ωk˜ake
−ik·x+˜a†
keik·x. Therefore, the ∂ ˜φhas an overall minus sign when compared to its counterpart ∂φ. This
2
The ER=EPR proposal
Black holes are solutions of General Relativity. Their peculiarity is the presence of an event horizon, which prevents anything, including light, that is inside the black hole to escape, there-fore making it black. They are characterized only by mass, angular momentum and electric charge.
The attempt to obtain a theory of quantum gravity is focused mostly on the study of black holes, since quantum effects become important in this situation, while in the vast majority of gravitational situations they can be neglected. However, black holes still represent an open issue for theoretical physics, since the presence of these quantum effects leads to apparent inconsistencies in the theory. Studying and understanding the origin of these paradoxes can help us moving closer to a theory of quantum gravity.
In the following chapter we will explain the ER=EPR proposal. First, we want to have a simple situation we can study. In order to do so, we will briefly review the AdS/CFT corre-spondence [2]. We will then talk about the BTZ black hole [5], a black hole solution in AdS2+1,
and some of its properties, for which the AdS/CFT correspondance has proven useful [6]. Since gravity in 2+1 dimensions does not present any propagating degree of freedom, i.e. gravitons, and its field theory dual is a two dimensional CFT, which has the advantage of having an infi-nite dimensional symmetry goup, it provides an accessible example to investigate the physics of black holes and has been intensively studied in the last twenty years.
The BTZ black hole will serve as the simplest example of a black hole we can study. We will finally introduce the information paradox and some of the solutions that have been proposed, such as the firewall [3] and the ER=EPR proposal [4].
2.1 AdS space and the AdS/CFT correspondence
Anti de Sitter space is one of the maximally symmetric spaces. This means that AdSd has
d(d+1)/2 killing vectors, the largest number possible for a d dimensional spacetime. Contrary to the other maximally symmetric spaces, it has a negative cosmological constant. AdSncan be
defined as an embedded hyperboloid of radius`in a n+1 dimensional flat space.
−X2−1−X20+X12+. . .+X2n−1= `2 (2.1)
The BTZ black hole 20
The metric can be expressed in global coordinates
ds2= `2(−cosh2ρdt2+dρ2+sinh2ρdΩd−2) (2.2)
with t∈ [0, 2π], and ρ∈R+. This choice of coordinates covers the whole AdS space.
The Poincar´e patch instead covers only half of AdS. The metric in this choice of coordinates is ds2= `2 z2(dz 2−dt2+d
∑
−2 i=1 dx2i). (2.3)Using the substitution z= u1 we find the metric
ds2 = `2 du 2 u2 −u 2(dt2+d
∑
−2 i=1 dx2i) . (2.4)The space has a conformal boundary at u = ∞ (or z = 0). This boundary is of vital interest for the AdS/CFT correspondence [2], which relates a theory of gravity living on AdSd+1 to
a conformal theory living on this boundary. More specifically, the AdS/CFT correspondence states that he R ddxφ 0(x)O(x)i CFT =Zbulk[φ(x, z)] φ(x,0)=φ0(x) . (2.5)
φ0(x)represents the boundary value of the field in the bulk φ(x, z). Both the right and left hand
sides of the last formula might present divergences, and in general it is necessary to carry out a renormalization procedure. See for example [17].
For our purposes, it will be enough to use the following formula. When taking a n-point function in the bulk and sending it to the boundary, one has the relation [18]
hO(x1). . . O(xn)iCFT =limz→ 0z
−n∆hφ(x
1, z). . . φ(xn, z)ibulk. (2.6)
The mass m of the field φ is related to the scaling dimension∆ of the CFT operator O by ∆= d 2+ r d2 4 +m 2`2. (2.7)
2.2 The BTZ black hole
The BTZ black hole can be obtained as a quotient of AdS2+1[19]. This turns out to be useful,
since some quantities, e.g. the geodesic length between two points, will be the same in the AdS and BTZ case.
We will focus only on the non-rotating chargeless solution. In this case the metric can be written as ds2 =−r 2−r2 ∗ `2 dt 2+ `2 r2−r2 ∗ dr2+r2dφ2. (2.8) The radius at which the event horizon is situated is r∗, while`is the radius of AdS. For
sim-u v
Figure 2.1: Kruskal diagram for the BTZ black hole. The thick lines represent the boundaries and the zigzag lines represent the future and past singularity.
C A
B
D
Figure 2.2: Penrose diagram for the BTZ black hole. A and C are the right and left exterior.
plicity, from here on we will set`to 1. The φ coordinate is periodically identified, φ =φ+2π.
The mass of the black hole M is related to the black hole radius as M = r2∗
8, and the temperature
of the black hole is β−1= r∗
2π.
It is possible to find the maximally extended solution of the BTZ space time by switching to Kruskal coordinates. The metric becomes [6]
ds2 =−(4dudv 1+uv)2+r 2 ∗ (1−uv)2 (1+uv)2dφ 2. (2.9)
The future and past singularities are located at uv = 1. We have two boundaries of AdS, located at uv = −1. In the form (2.8) the boundaries are located at r = ∞. In this limit the metric reduces (ignoring a conformal factor) to
ds2 ∼dt2+dφ2. (2.10) Therefore the CFTs on the boundaries are defined on R×S1.
The BTZ black hole 22
the black hole. A and C can be interpreted as two disconnected spaces, or as distant regions of the same space. It is possible to have a path that goes from region A to region C; this is called Einstein Rosen bridge, or wormhole. It would appear at first sight that this is a “shortcut” through space time, and could therefore violate causality. It is easy to see from figure2.2that no signal can travel from A to C; it is normally said that the length of the bridge increases so that no signal can possibly go through.
2.2.1 Propagators for the BTZ black hole
This discussion follows the one from [19].
In order to obtain the bulk to boundary propagator for the BTZ black hole, we can exploit the fact that it is a quotient of AdS3. Therefore we just need to add a sum over images to the
bulk to boundary propagator of the latter, to take into account the periodic identification of φ. Let us consider the bulk to boundary propagator for a scalar field of mass m in region A. The point x0 = (t, φ, r) = (x, r)is located in the right bulk, while the point z= (t0, φ0)is on the boundary of region A. Then the propagator will be
KAA(x, r; z)∼
∑
∞ n=−∞ − s r2−r2 ∗ r2 ∗ cosh r∗∆t+ r r∗ cosh r∗(∆φ+2πn) −2∆ (2.11) where∆=1+√1+m2,∆t=t−t0and∆φ=φ−φ0.If we now want to move one point to region C, it is sufficient to shift t → t+iβ/2. If the point x0is in region C, we have
KCA(x, r; z)∼
∑
∞ n=−∞ s r2−r2 ∗ r2 ∗ cosh r∗∆t+ r r∗cosh r∗(∆φ+2πn) −2∆ . (2.12)We can see that, while KAA might diverge for some values of r, ∆φ and ∆t, K
CA is always
non singular. This is consistent with the fact that, in general, propagators diverge on light like geodesics. There is no such geodesic that connects region 1 with region 3.
We can obtain the boundary to boundary propagator by sending the bulk point x to the boundary. According to the AdS/CFT correspondence, this will be obtained by the limit
P(x, z) =C lim
r→∞r
2∆K(x, r; z) (2.13)
with C being a r independent factor. Applying this limit we obtain
PAA(x, z)∼
∑
∞ n=−∞ −cosh r∗∆t+cosh r∗(∆φ+2πn) −2∆ PCA(x, z)∼∑
∞ n=−∞ cosh r∗∆t+cosh r∗(∆φ+2πn) −2∆ (2.14)u v
Figure 2.3: Constant t slice in Kruskal coordinates
decreasing towards the future. In this case,∆t is to be intended as ∆t= t+t0. The reason for this is that the ratio between the Kruskal coordinates u and v is only dependent on time (see [20] for the explicit form of u and v)
u
v = f(t) (2.15)
therefore a slice with t constant is represented by a straight line passing through the point u=v=0, as shown in figure2.3.
2.2.2 Holography for the BTZ black hole
It was shown in [6] that performing the path integral on the boundary CFT gives a state
|Ψi = √1
Z
∑
ie−βEi/2|ii
L⊗ |iiR. (2.16)
As seen in chapter 1, this is the thermofield state. The propagators obtained from the bulk, (2.14), are the same as the one obtained with the Schwinger Keldysh formalism in the boundary CFTs in section1.6.3for a field with scaling dimension (∆, ∆), if we include also a sum over images due to the periodic identification of the spatial coordinate. This is consistent with the claim (2.6). The BTZ black hole is dual to the TFD state.
The fact that the insertion of two operators on two independent and non interacting theories has a non vanishing correlator can be understood as the effect of the entangled state of the theories, or, equivalently, of the presence of the wormhole connecting the two theories.
We considered only the case with a non rotating and chargeless black hole. In the case of a conserved charge Q, the state (2.16) will also have a chemical potential µ
|Ψi = √1
Z
∑
i e−βEi/2−βµQi/2|E
Hawking radiation 24 I− I+ H+ H + I+ I− r =0 r = 0
Figure 2.4: Penrose diagram for a black hole originated by collapse. H represents the event horizon andI+andI−future and past null infinity.
2.3 Hawking radiation
So far, we have ignored quantum effects. When taking them into consideration on a black hole background, the situation becomes more complicated. Quantum effects on a curved spacetime are responsible for a black hole radiation, known as Hawking radiation [1]. This eventually leads to the information paradox.
We will consider a black hole formed by collapse in flat space, as originally studied by Hawking. This case can be generalized to that of the BTZ black hole. In the case of a black hole formed by collapse, the Penrose diagram reduces to the one shown in figure2.4, since the usual Schwarzschild metric is valid only in the region outside the collapsing matter.
The Schwarzschild metric in d dimensions is ds2 =− 1−2M r dt 2+ dr2 1−2Mr +r 2dΩ2 d−2, (2.18)
where dΩdis the metric on a d dimensional sphere. The regionsI−andI+are the past and
future null infinity. When we consider the limit r →∞, the metric reduces to the flat space one, so these regions are asymptotically flat.
Let us now consider a real scalar field φ, which we take to be massless for simplicity. The field satisfies the Klein Gordon equation, which reduces to the usual flat space Klein Gordon equationφ=0 at infinity. The mode expansion for the scalar field on regionI−is
φ=
∑
ωfωaω+ fω∗a †
ω (2.19)
where the coefficients fω solve the Klein Gordon equation. We consider the f coefficients to
contain only positive frequencies so that aωand a†ωare annihilation and creation operators on
the region considered. A massless field in the region outside the event horizon will be fully described by the choice of the coefficients fω.
we need to consider not only the regionI+, but also the event horizon. We can determine the field as φ=
∑
ω pωbω+p∗ωb † ω+qωcω+q∗ωc † ω. (2.20)The p modes represent the outgoing modes onI+and the q modes the ingoing on the event horizon. Again we restrict to positive frequencies only onI+, so that bω and b†ωare
annihila-tion and creaannihila-tion operator. The same cannot be done on the event horizon, since the splitting between positive and negative frequencies is possible when a timelike killing vector exists. On the horizon, the killing vector becomes null. This, however, is not relevant for our computation. Since φ can be determined by information on I− or on I+ and the event horizon, the coefficients f , p and q or the operators a, b and c are not independent of each other. They are related by some Bogoliubov transformation
pω =
∑
ω0 αωω0fω0+βωω0fω∗0 bω =∑
ω0 α∗ωω0aω0−β∗ωω0aω0 (2.21)and so on. The Bogoliubov coefficients satisfy the condition
∑
ω0
|αωω0|2− |βωω0|2 =1 (2.22)
which gives consistent commutation relations for the a and b operators.
Let us now consider the vacuum for the a operator. This is given by|0iinwith the condition aω|0iin =0 ∀ω. (2.23)
The outgoing vacuum |0ioutwill be given by the same condition, this time for the operator b.
However, in general,|0ioutwill not be the same as|0iin, which will not be annihilated by a b operator. Let us consider the b number operator Nb = ∑ωb
†
ωbω, and compute its expectation
value on the state|0iin. This will be given, using (2.21), by hNbiin =
∑
ωω0
|βωω0|2 (2.24)
and will not vanish in general.
The value of the βωω0coefficients can be computed (as in [1]) by considering the radial part
of the mode decomposition at past and future infinity. The modes onI+ can be of two kind. It could be a mode of energy ω onI−which is reflected by the Schwarzschild static potential and ends up onI+ with the same frequency ω. It could otherwise be a mode that scattered inside the collapsing matter, and ends up onI+with a different frequency ω0. The interesting effects are given by the second type of modes, for which the result is
The information paradox and the firewall proposal 26
where κ represents the surface gravity of the black hole, in this case 1/4M. Combining this with the condition (2.22), we obtain that
∑
ω0
|βωω0|2 =
1
e2πω/κ−1. (2.26)
This is the expectation value of b†
ωbω, the expected number of particles in every ω mode. It is a
Bose-Einstein distribution for a temperature of T= κ
2π = 1
8πM. (2.27)
Hawking radiation can also be pictured in a simple manner. It consists of two virtual parti-cles being created just outside the black hole horizon. There is the possibility that, before they annihilate each other, one of the two falls inside the black hole. The other particle will move away from the black hole. The sum of many similar processes gives rise to Hawking radiation. Since the time translation Killing vector becomes spacelike inside the horizon, the energy of the infalling quanta is considered to be negative. The black hole loses mass and eventually evaporates.
In the case of the BTZ black hole, the situation is slightly different. This black hole did not originate by collapse, and its Penrose diagram is the one shown in figure 2.2. Because of the Unruh effect [21], a distant static observer sees a non zero temperature for the black hole. Since the space is asymptotically AdS, the black hole is thermodynamically stable [22], and is eternal. AdS acts as a confining box, so all the particles emitted by Hawking radiation will eventually come back to the black hole. It is still possible to study the evaporation of a BTZ black hole by coupling it to sources.
2.4 The information paradox and the firewall proposal
The presence of the Hawking radiation has important consequences on the physics of black holes. Consider for example a black hole that originated from the collapse of a star, which is in an initial pure state. Particles emitted through Hawking radiation are entangled with particles inside of the black hole. After the black hole evaporates, we are left with a thermal density matrix, and it looks like we evolved from a pure to a mixed state1. This is not allowed
by quantum mechanics, since it would violate unitarity. The loss of information is also not consistent with the AdS/CFT correspondence, according to which the black hole is dual to a thermal theory which follows the rules of quantum mechanics. Unitarity in this latter theory implies that the black hole evolves unitarily as well.
Let us now introduce the concept of typical state, which will play an important role later on in the discussion. A typical state is a state for which the expectation value of simple observable
1It was argued by [23] that even though there could be small corrections to Hawking radiation, these will not be
enough to make this process unitary. However, in [24,25], the authors argue that the presence of small corrections becomes important once Page time is reached and eventually unitarizes Hawking radiation. This last construction presents a small violation of locality, which however cannot be detected within the limits of effective field theory.
quantities is close to the thermal expectation value. For example, one expects that almost all of the one point functions will be close to their thermal expectation value.
If we consider an evaporating system in a typical state, information will significantly start leaking out after half of the degrees of freedom of the system have already evaporated [26]. In the case of black holes, this time is called Page time. This happens because subsystems smaller than half of the system contain a negligible amount of information. The leaking out of information is possible thanks to the entanglement between the late and early radiation. This, however, presents a different problem. Let us call C a quantum of the early Hawking radiation, and B a quantum emitted in the late Hawking radiation, i.e. after the Page time. B has also a partner just inside the horizon, A.
Hawking radiation is a process which creates pairs of particles in a highly entangled pure state. Therefore, we can approximate the quantum B to be maximally entangled with A; due to the leaking out of information, B must be maximally entangled with C. We are facing a seri-ous paradox, since monogamy of entanglement, the principle for which one quantum system cannot be maximally entangled with two independent systems, is violated.
This explanation of the paradox is intuitively clear, but it is not very rigorous, since it does not take into account energy conservation; for example, the quanta A and B are in a state which is not maximally entangled, but rather thermally entangled. A more rigorous version of the paradox can be presented using the strong subadditivity of entropy [3]. Let us consider three quantum systems A, B and C and the total Hilbert space, H = HA⊗HB⊗HC. The
density matrix onH is ρABC, while ρAB is the reduced density matrix obtained by tracing out
HC, and so on. Si is the entropy associated with a density matrix, i.e. Si =−Tr(ρilog ρi). Then
the strong subadditivity of entropy states that
SAB+SBC ≥SABC+SB. (2.28)
Now let us consider again the evaporating black hole. If we assume that nothing unusual hap-pens when crossing the horizon (this assumption is often referred to as the no drama hypoth-esis), then B and A, which were created in a pure state, are still in a pure state. This implies SAB = 0 and SABC = SC. Besides, if the black hole is old, its entropy is decreasing, because
once the black hole is evaporated we will obtain a pure state. Therefore, SBC< SC. Finally, the
strong subadditivity of entropy reduces to SC >SB+SC, which is of course wrong, since B has
a thermal density matrix.
In the AMPS paper [3], it was argued that the following hypothesis cannot be valid at the same time:
1. the process of evaporation of a black hole obeys unitarity; 2. outside of the horizon, physics is approximately semiclassical;
3. the crossing of the horizon does not present any special feature for a freely falling ob-server (no drama hypothesis).
The ER=EPR proposal 28
It was proposed in the AMPS paper that the quanta A and B are not in an entangled state. This means giving up the no drama hypothesis, because an observer entering an old enough black hole would encounter an energetic firewall. This proposal, however, violates Einstein’s equivalence principle, because a local free-falling observer would see the black hole horizon as a special place, even though this should be locally flat instead.
It is also argued by AMPS that, while describing the evaporation of the black hole is prob-lematic only after the Page time, the firewall is present even for “young” black holes. The authors claim that the interesting time scale for this process is the scrambling time, i.e. the time necessary for thermalization to occur. After the scrambling time any subset of the degrees of freedom reaches a maximally mixed state, and therefore can be considered “old”. Besides, the firewall is a feature of the black hole and it should not depend on the entanglement with an external system.
2.5 The ER=EPR proposal
A different solution to the paradox was put forward in [4] by Maldacena and Susskind. Ac-cording to the authors, there is a correspondence between entanglement and Einsten-Rosen bridges. The case of the BTZ black hole is one example for which the TFD state is dual to a wormhole, but the authors suppose that this correspondence is more general. They also argue that simpler systems, such as two entangled qubits, are connected by some bridge; however, this bridge might be highly quantum, therefore not presenting a smooth geometry. Given our present understanding of non classical geometries, it is not possible to study these quantum wormholes. It appears that such a task will be feasible only with a fully working theory of quantum gravity. This theory will already describe the process of evaporation of a black hole without encountering paradoxes, therefore the ER=EPR might not represent the correct path to follow at the moment.
The authors consider an Einstein-Rosen bridge of the kind shown in figure2.2. The left and right exterior areas A and C are considered distant regions of the same spacetime, rather than two disconnected regions. The microstates of these black holes are entangled in EPR pairs. Wherever a one sided black hole is considered, e.g. a black hole which forms after the collapse of a star, we can consider Hawking radiation to play the role of the second black hole. An old one sided black hole becomes effectively a two sided black hole. An observer that collected all the radiation up to the Page time could, using a powerful enough quantum computer, make it collapse into a black hole. This new black hole would be entangled with the other black hole, and they would be connected by a ER bridge2. As explained before, it is impossible at the mo-ment to investigate quantum wormholes. However, the previous picture is consistent with the presence of these quantum wormholes between quanta of the Hawking radiation and quanta inside the black hole. The semiclassical wormhole obtained when collapsing the Hawking ra-diation is created by merging together all the quantum wormholes connecting the black hole
2In [24] it was instead proposed that the splitting of the degrees of freedom is due to coarse graining. The coarse
A B t −t A0 t A00
Figure 2.5: The mode B reaches the boundary at time t, while A0 at time−t. A hits the singu-larity.
to its radiation.
The simplest way to see how this proposal differs from the AMPS proposal is to imagine a condensed matter system in a laboratory prepared in the thermofield state, with a gravity dual represented by a ER bridge. An observer, Alice, is in the laboratory, and can move particles be-tween the right and left theories. A qubit B is traveling in the right bulk theory, and its infalling partner A is inside the black hole, as depicted in figure2.5. They are entangled. Eventually, B will reach the boundary at a time t and A will hit the singularity. However, we know from the thermofield double state that B is entangled also with a mode in the left boundary at time−t3, A0. It is possible to see from the Penrose diagram that the qubit A is obtained by evolving A0 in the bulk, so the fact that B is entangled with both A and A0is not a problem, since A and A0 are not independent. A mode inside the black hole can be influenced by signals traveling from both boundaries. The fact that A and A0 are not independent is schematically represented by
[A, A0]6=0. (2.29)
A0 is living on the left boundary, so we can make it evolve in time from time−t to t, and we will obtain the mode A00. Even though they are located at a great distance, A00will not be independent of A, so
[A, A00]6=0. (2.30)
We can now imagine Alice distilling the qubit A0 and taking it to the second boundary, where she gives it to an observer Bob. Bob enters the black hole to measure the entanglement between A and B. Bob will hit an energetic particle O just inside the horizon. This particle is there, however, because Alice corrupted the mode A when making a measurement on A0,
3The fact that time is defined as decreasing towards the future for the left boundary does not play a role here,
The ER=EPR proposal 30 B A0 O A0 B O A0
Figure 2.6: Left: Alice distills a qubit A0and brings it to a boundary. Right: Alice moves in the laboratory (red line) and takes the qubit to the other boundary, where she gives it to Bob. Bob jumps into the black hole and finds an energetic particle just behind the horizon.
since they are not independent. The same reasoning can be applied if Alice distills A00, since
[A, A00]6=0.
Bob, the observer jumping into a black hole, finds an energetic particle the moment in which he enters the black hole; however, this is not a firewall, in the sense of AMPS, since this particle has been created in the mode A only by the action of Alice. The AMPS proposal concludes instead that all possible modes present energetic particles, and these are not created by a mea-surement of an observer.
Let us now drop the experiment in the lab and focus on a black hole. Alice is an observer that collects all the Hawking radiation from a one sided black hole, and collapses it into another black hole, which will be entangled with the initial one. The two will also be located in the same region of space, so it is possible to travel from one to the other. Then Alice distills a qubit C from her black hole and travels to the initial black hole in time to measure the emitted qubit B, which will be entangled with C. After measuring the entanglement between the two particles, she jumps into the black hole. Because of her measurement on C, Alice’s horizon crossing will not be drama free: she will encounter an energetic particle, which has been created by the distillation of C. This particle is present only for the mode A, not for the other modes, and comes from the other side of the wormhole.
It is not necessary for Alice to collapse the radiation into a black hole. The qubits emitted from the black hole form EPR pairs with qubits inside the black hole: according to the proposal, they are also connected by some kind of ER bridge. Distilling one qubit of the early radiation corrupts the entanglement between A and B, and an observer entering the black hole will find an energetic particle; it is the ER bridge which is responsible for this action at a distance.
Therefore, according to the proposal, the presence of a firewall will not be only determined by the black hole itself, but it will depend also on the system with which the black hole is entangled.
2.5.1 Reactions to the ER=EPR proposal
The proposal by Maldacena and Susskind is elegant, but it makes conjectures, i.e. that any two entangled systems are connected by a possibly quantum wormhole, which cannot be verified until a working theory of quantum gravity exists. The work on the ER=EPR proposal concerns the presence of semiclassical ER bridges only.
Evidence against the ER=EPR proposal was first provided by Marolf and Polchinski [7], two authors of the AMPS paper. They drew the conclusion that, in general, typical states are not connected by a wormhole. They consider the strongly coupled boundary CFT to be a chaotic system, and they use the eigenvalue thermalization hypothesis (ETH)
hα|A|βi =Aαβ =A(Eαβ)δαβ+e−S
(Eαβ)fA(E
α, Eβ)RαβA (2.31)
with Eαβ = (Eα+Eβ)/2. A and fA are smooth functions, and RαβA varies stochastically. Its
average is zero, and its variance is
hRαβARBγδi =δαδδβγσ AB(E
α, Eβ) (2.32)
with σAB again a smooth function. The authors show that the expectation value of the prod-uct of operators on different theories on a general state|Ψicannot be larger than e−S. On the contrary, if the state |Ψiis the thermofield state, the correlation is not suppressed. The au-thors claim that whenever a correlator is suppressed by a factor depending on the number of states, then the two theories are not connected by a wormhole. This happens because the ge-ometrical construction is not sensitive to the details of the spectrum of the field theory, since it should arise from the coarse graining of the latter. Besides, under coarse graining a typical state appears as thermal; we would expect the wormhole corresponding to a typical state to be similar to the one of the TFD state, which presents no e−S suppression. This does not happen and it would appear that, for general states, entanglement does not imply the presence of a semiclassical ER.
A similar result was obtained in [8], using a random matrix approach instead of ETH. In particular, it was shown that, while a unitary twisting of the thermofield double concerning only one of the theories leaves the reduced density matrix of the second theory unchanged, this affects correlators between the two theories. Under such a transformation, this two sided correlator is smaller in average than the one sided correlator by a factor of e−S. The authors conclude that these low correlations imply the non existence of a wormhole. Besides, it is possible to have high correlations between two theories without having a wormhole connecting them, as in the case of a temperature lower than the Hawking-Page transition, for which the thermofield state is dual to two copies of thermal AdS. Combining these results, it appears that the ER=EPR proposal is inconsistent. More on this work is presented in chapter4.
However, in [9] the effects of addition of particles on the boundary of the BTZ black hole were studied, and in [10] this was used by the same authors to create long smooth wormholes. These semiclassical wormholes can be as long as one might like, provided that quantum
